### On the total length of external branches for beta-coalescents

#### Abstract

In this paper we consider the beta(2 - α, α)-coalescents with 1 < α < 2 and study the moments of external branches, in particular, the total external branch length Lext(n) of an initial sample of n individuals. For this class of coalescents, it has been proved that nα-1T(n)D T, where T(n) is the length of an external branch chosen at random and T is a known nonnegative random variable. For beta(2 - α, α)-coalescents with 1 < α < 2, we obtain limn→+∞n3α-5 E{(Lext(n) - n2-αE{T})2} = ((α - 1)Γ(α + 1))2Γ(4 - α) / ((3 - α)Γ(4 - 2α)).

#### Article information

Source
Adv. in Appl. Probab., Volume 47, Number 3 (2015), 693-714.

Dates
First available in Project Euclid: 8 October 2015

https://projecteuclid.org/euclid.aap/1444308878

Digital Object Identifier
doi:10.1239/aap/1444308878

Mathematical Reviews number (MathSciNet)
MR3406604

Zentralblatt MATH identifier
1328.60195

#### Citation

Dhersin, Jean-Stéphane; Yuan, Linglong. On the total length of external branches for beta-coalescents. Adv. in Appl. Probab. 47 (2015), no. 3, 693--714. doi:10.1239/aap/1444308878. https://projecteuclid.org/euclid.aap/1444308878

#### References

• Árnason, E. (2004). Mitochondrial cytochrome $b$ DNA variation in the high-fecundity Atlantic cod: Trans-Atlantic clines and shallow gene genealogy. Genetics 166, 1871–1885.
• Berestycki, J., Berestycki, N. and Limic, V. (2014). A small-time coupling between $\Lambda$-coalescents and branching processes. Ann. Appl. Prob. 24, 449–475.
• Berestycki, J., Berestycki, N. and Limic, V. (2014). Asymptotic sampling formulae for $\Lambda$-coalescents. Ann. Inst. H. Poincaré Prob. Statist. 50, 715–731.
• Berestycki, J., Berestycki, N. and Schweinsberg, J. (2007). Beta-coalescents and continuous stable random trees. Ann. Prob. 35, 1835–1887.
• Berestycki, N. (2009). Recent Progress in Coalescent Theory (Ensaios Matemáticos 16). Sociedade Brasileira de Mathemática, Rio de Janeiro.
• Birkner, M. et al. (2005). Alpha-stable branching and beta-coalescents. Electron. J. Prob. 10, 303–325.
• Blum, M. G. B. and François, O. (2005). Minimal clade size and external branch length under the neutral coalescent. Adv. Appl. Prob. 37, 647–662.
• Bolthausen, E. and Sznitman, A.-S. (1998). On Ruelle's probability cascades and an abstract cavity method. Commun. Math. Phys. 197, 247–276.
• Boom, J. D. G., Boulding, E. G. and Beckenbach, A. T. (1994). Mitochondrial DNA variation in introduced populations of Pacific Oyster, Crassostrea gigas, in British Columbia. Canad. J. Fisheries Aquatic Sci. 51, 1608–1614.
• Bovier, A. and Kurkova, I. (2007). Much ado about Derrida's GREM. In Spin Glasses (Lecture Notes Math. 1900), Springer, Berlin, pp. 81–115.
• Breiman, L. (1992). Probability (Classics Appl. Math. 7). SIAM, Philadelphia, PA.
• Caliebe, A., Neininger, R., Krawczak, M. and Rösler, U. (2007). On the length distribution of external branches in coalescence trees: genetic diversity within species. Theoret. Pop. Biol. 72, 245–252.
• Delmas, J.-F., Dhersin, J.-S. and Siri-Jégousse, A. (2008). Asymptotic results on the length of coalescent trees. Ann. Appl. Prob. 18, 997–1025.
• Dhersin, J.-S. and Möhle, M. (2013). On the external branches of coalescents with multiple collisions. Electron. J. Prob. 18, 11pp.
• Dhersin, J.-S. and Yuan, L. (2012). Asympotic behavior of the total length of external branches for beta-coalescents. Preprint. Available at http://arxiv.org/abs/1202.5859.
• Dhersin, J.-S., Freund, F., Siri-Jégousse, A. and Yuan, L. (2013). On the length of an external branch in the beta-coalescent. Stoch. Process. Appl. 123, 1691–1715.
• Drmota, M., Iksanov, A., Moehle, M. and Roesler, U. (2007). Asymptotic results concerning the total branch length of the Bolthausen–Sznitman coalescent. Stoch. Process. Appl. 117, 1404–1421.
• Durrett, R. (2008). Probability Models for DNA Sequence Evolution, 2nd edn. Springer, New York.
• Eldon, B. and Wakeley, J. (2006). Coalescent processes when the distribution of offspring number among individuals is highly skewed. Genetics 172, 2621–2633.
• Foucart, C. and Hénard, O. (2013). Stable continuous-state branching processes with immigration and beta–Fleming–Viot processes with immigration. Electron. J. Prob. 18, 21pp.
• Freund, F. and Möhle, M. (2009). On the time back to the most recent common ancestor and the external branch length of the Bolthausen–Sznitman coalescent. Markov Process. Relat. Fields 15, 387–416.
• Fu, Y. X. and Li, W. H. (1993). Statistical tests of neutrality of mutations. Genetics 133, 693–709.
• Gnedin, A. and Yakubovich, Y. (2007). On the number of collisions in $\Lambda$-coalescents. Electron. J. Prob. 12, 1547–1567.
• Gnedin, A., Iksanov, A. and Möhle, M. (2008). On asymptotics of exchangeable coalescents with multiple collisions. J. Appl. Prob. 45, 1186–1195.
• Goldschmidt, C. and Martin, J. B. (2005). Random recursive trees and the Bolthausen–Sznitman coalescent. Electron. J. Prob. 10, 718–745.
• Hedgecock, D. (1994). Does variance in reproductive success limit effective population sizes of marine organisms? In Genetics and Evolution of Aquatic Organisms, Chapman & Hall, London, pp. 122–134.
• Janson, S. and Kersting, G. (2011). On the total external length of the Kingman coalescent. Electron. J. Prob. 16, 2203–2218.
• Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.
• Kersting, G. (2012). The asymptotic distribution of the length of beta-coalescent trees. Ann. Appl. Prob. 22, 2086–2107.
• Kersting, G., Pardo, J. C. and Siri-Jégousse, A. (2014). Total internal and external lengths of the Bolthausen–Sznitman coalescent. In Celebrating 50 years of the Applied Probability Trust (J. Appl. Prob. Spec. Vol. 51A), Applied Probability Trust, Sheffield, pp. 73–86.
• Kersting, G., Stanciu, I. and Wakolbinger, A. (2014). The total external branch length of beta-coalescents. Combin. Prob. Comput. 23, 1010–1027
• Kimura, M. (1969). The number of heterozygous nucleotide sites maintained in a finite population due to steady flux of mutations. Genetics 61, 893–903.
• Kingman, J. F. C. (1982). The coalescent. Stoch. Process. Appl. 13, 235–248.
• Kingman, J. F. C. (2000). Origins of the coalescent: 1974–1982. Genetics 156, 1461–1463.
• Marynych, O. (2011). Stochastic recurrences and their applications to the analysis of partition-valued processes. Doctoral Thesis, Utrecht University.
• Möhle, M. (2006). On the number of segregating sites for populations with large family sizes. Adv. Appl. Prob. 38, 750–767.
• Möhle, M. (2010). Asymptotic results for coalescent processes without proper frequencies and applications to the two-parameter Poisson–Dirichlet coalescent. Stoch. Process. Appl. 120, 2159–2173.
• Pitman, J. (1999). Coalescents with multiple collisions. Ann. Prob. 27, 1870–1902.
• Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Prob. 36, 1116–1125.
• Schweinsberg, J. (2003). Coalescent processes obtained from supercritical Galton–Watson processes. Stoch. Process. Appl. 106, 107–139.
• Yuan, L. (2014). On the measure division construction of $\Lambda$-coalescents. Markov Process. Relat. Fields 20, 229–264.